MATH 1130: Show Using an Honest Hilbert Proof that ⊒𝐴∧(𝐴∨𝐡)≑𝐴: Discrete Mathematics 1 Assignment, DC, Canada

University Douglas College (DC)
Subject MATH 1130: Discrete Mathematics 1

Question 1:

Show using an honest Hilbert proof that ⊒𝐴∧(𝐴∨𝐡)≑𝐴.

For this question, your answer must be a Hilbert proof according to Definition 1.4.5, no shortcuts (including meta theorems or absolute theorems from class/book) are allowed.

Question 2:

(a) Prove that for any wffs 𝐴,𝐡,𝐢, D and any propositional variable 𝑝,

𝐴→(𝐡≑𝐢)βŠ’π΄β†’(𝐷[𝑝:=𝐡]≑𝐷[𝑝:=𝐢])

(Hint: Using the deduction theorem or Post’s theorem makes this easier, although there are other proofs).

(b) Is it true that 𝐴→(𝐡→𝐢)βŠ’π΄β†’(𝐷[𝑝:=𝐡]→𝐷[𝑝:=𝐢])? If yes, give a proof of this, and if no, find a counterexample.

Question 3:

Consider the stringΒ ((((π‘βˆ¨π‘Ÿ)β‰‘π‘ž)β†’((π‘Ÿβˆ§(¬𝑝))β‰‘π‘Ÿ))β†’π‘Ÿ)

(a) Write a formula-calculation to show this string is a wff.

(b) Is this formula a tautology? Explain why or why not.

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Question 4:

(a) Give an example of a well-formed formula 𝐴 so that we do NOT have ⊒¬𝐴. Explain why your answer works (using soundness might help).

(b) What is wrong with the following equational proof of ⊒¬𝐴?

¬𝐴

⇔⇔ (Β¬-introduction axiom)

𝐴≑βŠ₯

⇔⇔ (Leibniz from prev. line, “C-part” 𝑝≑βŠ₯)

βŠ₯≑βŠ₯

⇔⇔ (⊀ vs.Β βŠ₯)

⊀

Question 5:

Use the method we discussed from the “Weak Post’s Theorem” notes to show:

⊒((π‘βˆ¨Β¬π‘žβˆ¨π‘Ÿ)∧(π‘β†’π‘Ÿ))β†’(π‘žβ†’π‘Ÿ)

Question 6:

Write an equational-style proof for the following: π‘Ÿβ†’(π‘βˆ¨π‘ž)⊒(π‘Ÿβˆ¨(π‘βˆ¨π‘ž))β†’(π‘βˆ¨π‘ž).

Question 7:

Give proof to show

⊒((π‘β†’π‘ž)∧(π‘β†’π‘Ÿ))β†’(𝑝→(π‘žβˆ§π‘Ÿ))

(Use equational or Hilbert proof styles. Post’s theorem is not allowed).

Question 8:

What is wrong with the following equational proof of ⊒(𝐴→𝐢)≑𝐴?

𝐴→𝐢

⇔⇔ (implication axiom)

π΄βˆ¨πΆβ‰‘πΆ

⇔⇔ (Leibniz+redundant true; “C-part”Β π΄βˆ¨π‘)

𝐴∨⊀

⇔⇔ (∨-identity)

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